To an extent. With regular development times, shadows tend not to be developed out entirely. Which makes sense as otherwise push processing would have no effect.
If your shadows are not developed to where you desire them, you are underexposing your film. Solution: shoot at a different E.I.
If your highlights are too dense to print well, you are overdeveloping. Solution: shorten development time
If your highlights are too weak, then vice-versa of the above.
All this applies to one scene with one subject brightness range (SBR) and a photographer who has a precise idea of how shadows should be rendered.
For the commonly-used "one development time fits all" approach, one must find an E.I. and a developing time that works for the great majority of photos made, allowing one to use whatever contrast adjustments are available at the printing stage to compensate for any variances.
For those who use the Zone System or something similar, having different development times for scenes with different SBRs is the norm. E.I. is typically adjusted by small amounts for changes in development time. A "Zonie" would never talk about "overexposing and underdeveloping," rather about "contraction development" and "supporting the shadows." The reduced development time for a scene with a large SBR is just SOP, and a small increase in exposure to compensate for the reduced development time is given to ensure the shadows are rendered where the photographer wants them to be.
For all of the above, it is primarily exposure (i.e., effective film speed) that determines how the shadows are reproduced; development time has a relatively small effect on film speed.
"Pushing," as the term is commonly used, is an intentional underexposure (for whatever reason: low light, etc.) and then compensatory overdevelopment to get a negative that will print better. Such a negative, by definition, has lost shadow detail and has an expanded contrast range for what's left; shadows are blank black, highlights can be gritty. It's a look that some like and intentionally strive for. Note, however, that the unexposed shadows never get somehow magically restored by extending the development time. Pushing works by sacrificing the shadows and expanding the contrast gradient for what's left.
"Pulling," i.e., blindly overexposing and underdeveloping is a rather imprecise short-cut to try and get a better negative, especially when one works in contrasty lighting conditions. Better would be doing some tests to find a good workable personal E.I. and a development time or two that worked well. Certainly, "overexpose by a stop and cut developing time by 25%" may be a good starting point, but careful workers who are interested in excellence will refine this.
Best,
Doremus